# Complex number calculator

There are 24 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains roots of complex numbers or powers to 1/n.

#### z1 = ((-5i)^(1/8))*(8^(1/3)) = 2.3986959-0.4771303i = 2.4456891 × ei -0.1963495 = 2.4456891 × ei (-0.0625) πCalculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei -0.1963495 = 1.2228445 × ei (-0.0625) π = 1.1993479-0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * 2 = 2.3986959-0.4771303i
The result z1
Rectangular form:
z = 2.3986959-0.4771303i

Angle notation (phasor):
z = 2.4456891 ∠ -11°15'

Polar form:
z = 2.4456891 × (cos (-11°15') + i sin (-11°15'))

Exponential form:
z = 2.4456891 × ei -0.1963495 = 2.4456891 × ei (-0.0625) π

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.1963495 rad = -11.25° = -11°15' = -0.0625π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.3986959-0.4771303i
Real part: x = Re z = 2.399
Imaginary part: y = Im z = -0.47713027

#### z2 = ((-5i)^(1/8))*(8^(1/3)) = 2.0335162+1.3587521i = 2.4456891 × ei 3π/16Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 3π/16 = 1.0167581+0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * 2 = 2.0335162+1.3587521i
The result z2
Rectangular form:
z = 2.0335162+1.3587521i

Angle notation (phasor):
z = 2.4456891 ∠ 33°45'

Polar form:
z = 2.4456891 × (cos 33°45' + i sin 33°45')

Exponential form:
z = 2.4456891 × ei 0.5890486 = 2.4456891 × ei 3π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.5890486 rad = 33.75° = 33°45' = 0.1875π = 3π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.0335162+1.3587521i
Real part: x = Re z = 2.034
Imaginary part: y = Im z = 1.35875206

#### z3 = ((-5i)^(1/8))*(8^(1/3)) = 0.4771303+2.3986959i = 2.4456891 × ei 7π/16Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 7π/16 = 0.2385651+1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * 2 = 0.4771303+2.3986959i
The result z3
Rectangular form:
z = 0.4771303+2.3986959i

Angle notation (phasor):
z = 2.4456891 ∠ 78°45'

Polar form:
z = 2.4456891 × (cos 78°45' + i sin 78°45')

Exponential form:
z = 2.4456891 × ei 1.3744468 = 2.4456891 × ei 7π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.3744468 rad = 78.75° = 78°45' = 0.4375π = 7π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.4771303+2.3986959i
Real part: x = Re z = 0.477
Imaginary part: y = Im z = 2.39869586

#### z4 = ((-5i)^(1/8))*(8^(1/3)) = -1.3587521+2.0335162i = 2.4456891 × ei 11π/16Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 11π/16 = -0.679376+1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * 2 = -1.3587521+2.0335162i
The result z4
Rectangular form:
z = -1.3587521+2.0335162i

Angle notation (phasor):
z = 2.4456891 ∠ 123°45'

Polar form:
z = 2.4456891 × (cos 123°45' + i sin 123°45')

Exponential form:
z = 2.4456891 × ei 2.1598449 = 2.4456891 × ei 11π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.1598449 rad = 123.75° = 123°45' = 0.6875π = 11π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.3587521+2.0335162i
Real part: x = Re z = -1.359
Imaginary part: y = Im z = 2.03351616

#### z5 = ((-5i)^(1/8))*(8^(1/3)) = -2.3986959+0.4771303i = 2.4456891 × ei 15π/16Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 15π/16 = -1.1993479+0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * 2 = -2.3986959+0.4771303i
The result z5
Rectangular form:
z = -2.3986959+0.4771303i

Angle notation (phasor):
z = 2.4456891 ∠ 168°45'

Polar form:
z = 2.4456891 × (cos 168°45' + i sin 168°45')

Exponential form:
z = 2.4456891 × ei 2.9452431 = 2.4456891 × ei 15π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.9452431 rad = 168.75° = 168°45' = 0.9375π = 15π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.3986959+0.4771303i
Real part: x = Re z = -2.399
Imaginary part: y = Im z = 0.47713027

#### z6 = ((-5i)^(1/8))*(8^(1/3)) = -2.0335162-1.3587521i = 2.4456891 × ei (-13π/16)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-13π/16) = -1.0167581-0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * 2 = -2.0335162-1.3587521i
The result z6
Rectangular form:
z = -2.0335162-1.3587521i

Angle notation (phasor):
z = 2.4456891 ∠ -146°15'

Polar form:
z = 2.4456891 × (cos (-146°15') + i sin (-146°15'))

Exponential form:
z = 2.4456891 × ei -2.552544 = 2.4456891 × ei (-13π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.552544 rad = -146.25° = -146°15' = -0.8125π = -13π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.0335162-1.3587521i
Real part: x = Re z = -2.034
Imaginary part: y = Im z = -1.35875206

#### z7 = ((-5i)^(1/8))*(8^(1/3)) = -0.4771303-2.3986959i = 2.4456891 × ei (-9π/16)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-9π/16) = -0.2385651-1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * 2 = -0.4771303-2.3986959i
The result z7
Rectangular form:
z = -0.4771303-2.3986959i

Angle notation (phasor):
z = 2.4456891 ∠ -101°15'

Polar form:
z = 2.4456891 × (cos (-101°15') + i sin (-101°15'))

Exponential form:
z = 2.4456891 × ei -1.7671459 = 2.4456891 × ei (-9π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.7671459 rad = -101.25° = -101°15' = -0.5625π = -9π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.4771303-2.3986959i
Real part: x = Re z = -0.477
Imaginary part: y = Im z = -2.39869586

#### z8 = ((-5i)^(1/8))*(8^(1/3)) = 1.3587521-2.0335162i = 2.4456891 × ei (-5π/16)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-5π/16) = 0.679376-1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = 2
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * 2 = 1.3587521-2.0335162i
The result z8
Rectangular form:
z = 1.3587521-2.0335162i

Angle notation (phasor):
z = 2.4456891 ∠ -56°15'

Polar form:
z = 2.4456891 × (cos (-56°15') + i sin (-56°15'))

Exponential form:
z = 2.4456891 × ei -0.9817477 = 2.4456891 × ei (-5π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.9817477 rad = -56.25° = -56°15' = -0.3125π = -5π/16 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.3587521-2.0335162i
Real part: x = Re z = 1.359
Imaginary part: y = Im z = -2.03351616

#### z9 = ((-5i)^(1/8))*(8^(1/3)) = -0.786141+2.3158967i = 2.4456891 × ei 29π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei -0.1963495 = 1.2228445 × ei (-0.0625) π = 1.1993479-0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * (-1+1.7320508i) = 1.19934793 * (-1) + 1.19934793 * 1.7320508076i + (-0.2385651361i) * (-1) + (-0.2385651361i) * 1.7320508076i = -1.19934793+2.07733155i+0.23856514i-0.41320694i2 = -1.19934793+2.07733155i+0.23856514i+0.41320694 = -1.19934793 + 0.41320694 +i(2.07733155 + 0.23856514) = -0.786141+2.3158967i
alternative steps
1.2228445 × ei -0.1963495 = 1.2228445 × ei (-0.0625) π × 2 × ei 2π/3 = 1.2228445 × 2 × ei ((-0.0625)+2π/3) = 2.4456891 × ei 29π/48 =
The result z9
Rectangular form:
z = -0.786141+2.3158967i

Angle notation (phasor):
z = 2.4456891 ∠ 108°45'

Polar form:
z = 2.4456891 × (cos 108°45' + i sin 108°45')

Exponential form:
z = 2.4456891 × ei 1.8980456 = 2.4456891 × ei 29π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.8980456 rad = 108.75° = 108°45' = 0.6041667π = 29π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.786141+2.3158967i
Real part: x = Re z = -0.786
Imaginary part: y = Im z = 2.31589669

#### z10 = ((-5i)^(1/8))*(8^(1/3)) = -2.1934719+1.0817006i = 2.4456891 × ei 41π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 3π/16 = 1.0167581+0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * (-1+1.7320508i) = 1.01675807982 * (-1) + 1.01675807982 * 1.7320508076i + 0.679376028827i * (-1) + 0.679376028827i * 1.7320508076i = -1.01675808+1.76107665i-0.67937603i+1.1767138i2 = -1.01675808+1.76107665i-0.67937603i-1.1767138 = -1.01675808 - 1.1767138 +i(1.76107665 - 0.67937603) = -2.1934719+1.0817006i
alternative steps
1.2228445 × ei 3π/16 × 2 × ei 2π/3 = 1.2228445 × 2 × ei (3π/16+2π/3) = 2.4456891 × ei 41π/48 =
The result z10
Rectangular form:
z = -2.1934719+1.0817006i

Angle notation (phasor):
z = 2.4456891 ∠ 153°45'

Polar form:
z = 2.4456891 × (cos 153°45' + i sin 153°45')

Exponential form:
z = 2.4456891 × ei 2.6834437 = 2.4456891 × ei 41π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.6834437 rad = 153.75° = 153°45' = 0.8541667π = 41π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.1934719+1.0817006i
Real part: x = Re z = -2.193
Imaginary part: y = Im z = 1.08170062

#### z11 = ((-5i)^(1/8))*(8^(1/3)) = -2.3158967-0.786141i = 2.4456891 × ei (-43π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 7π/16 = 0.2385651+1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * (-1+1.7320508i) = 0.2385651361 * (-1) + 0.2385651361 * 1.7320508076i + 1.19934793i * (-1) + 1.19934793i * 1.7320508076i = -0.23856514+0.41320694i-1.19934793i+2.07733155i2 = -0.23856514+0.41320694i-1.19934793i-2.07733155 = -0.23856514 - 2.07733155 +i(0.41320694 - 1.19934793) = -2.3158967-0.786141i
alternative steps
1.2228445 × ei 7π/16 × 2 × ei 2π/3 = 1.2228445 × 2 × ei (7π/16+2π/3) = 2.4456891 × ei (-43π/48) =
The result z11
Rectangular form:
z = -2.3158967-0.786141i

Angle notation (phasor):
z = 2.4456891 ∠ -161°15'

Polar form:
z = 2.4456891 × (cos (-161°15') + i sin (-161°15'))

Exponential form:
z = 2.4456891 × ei -2.8143434 = 2.4456891 × ei (-43π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.8143434 rad = -161.25° = -161°15' = -0.8958333π = -43π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.3158967-0.786141i
Real part: x = Re z = -2.316
Imaginary part: y = Im z = -0.78614099

#### z12 = ((-5i)^(1/8))*(8^(1/3)) = -1.0817006-2.1934719i = 2.4456891 × ei (-31π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 11π/16 = -0.679376+1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * (-1+1.7320508i) = -0.679376028827 * (-1) + (-0.679376028827) * 1.7320508076i + 1.01675807982i * (-1) + 1.01675807982i * 1.7320508076i = 0.67937603-1.1767138i-1.01675808i+1.76107665i2 = 0.67937603-1.1767138i-1.01675808i-1.76107665 = 0.67937603 - 1.76107665 +i(-1.1767138 - 1.01675808) = -1.0817006-2.1934719i
alternative steps
1.2228445 × ei 11π/16 × 2 × ei 2π/3 = 1.2228445 × 2 × ei (11π/16+2π/3) = 2.4456891 × ei (-31π/48) =
The result z12
Rectangular form:
z = -1.0817006-2.1934719i

Angle notation (phasor):
z = 2.4456891 ∠ -116°15'

Polar form:
z = 2.4456891 × (cos (-116°15') + i sin (-116°15'))

Exponential form:
z = 2.4456891 × ei -2.0289453 = 2.4456891 × ei (-31π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.0289453 rad = -116.25° = -116°15' = -0.6458333π = -31π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.0817006-2.1934719i
Real part: x = Re z = -1.082
Imaginary part: y = Im z = -2.19347188

#### z13 = ((-5i)^(1/8))*(8^(1/3)) = 0.786141-2.3158967i = 2.4456891 × ei (-19π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 15π/16 = -1.1993479+0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * (-1+1.7320508i) = -1.19934793 * (-1) + (-1.19934793) * 1.7320508076i + 0.2385651361i * (-1) + 0.2385651361i * 1.7320508076i = 1.19934793-2.07733155i-0.23856514i+0.41320694i2 = 1.19934793-2.07733155i-0.23856514i-0.41320694 = 1.19934793 - 0.41320694 +i(-2.07733155 - 0.23856514) = 0.786141-2.3158967i
alternative steps
1.2228445 × ei 15π/16 × 2 × ei 2π/3 = 1.2228445 × 2 × ei (15π/16+2π/3) = 2.4456891 × ei (-19π/48) =
The result z13
Rectangular form:
z = 0.786141-2.3158967i

Angle notation (phasor):
z = 2.4456891 ∠ -71°15'

Polar form:
z = 2.4456891 × (cos (-71°15') + i sin (-71°15'))

Exponential form:
z = 2.4456891 × ei -1.2435471 = 2.4456891 × ei (-19π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.2435471 rad = -71.25° = -71°15' = -0.3958333π = -19π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.786141-2.3158967i
Real part: x = Re z = 0.786
Imaginary part: y = Im z = -2.31589669

#### z14 = ((-5i)^(1/8))*(8^(1/3)) = 2.1934719-1.0817006i = 2.4456891 × ei (-7π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-13π/16) = -1.0167581-0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * (-1+1.7320508i) = -1.01675807982 * (-1) + (-1.01675807982) * 1.7320508076i + (-0.679376028827i) * (-1) + (-0.679376028827i) * 1.7320508076i = 1.01675808-1.76107665i+0.67937603i-1.1767138i2 = 1.01675808-1.76107665i+0.67937603i+1.1767138 = 1.01675808 + 1.1767138 +i(-1.76107665 + 0.67937603) = 2.1934719-1.0817006i
alternative steps
1.2228445 × ei (-13π/16) × 2 × ei 2π/3 = 1.2228445 × 2 × ei ((-13π/16)+2π/3) = 2.4456891 × ei (-7π/48) =
The result z14
Rectangular form:
z = 2.1934719-1.0817006i

Angle notation (phasor):
z = 2.4456891 ∠ -26°15'

Polar form:
z = 2.4456891 × (cos (-26°15') + i sin (-26°15'))

Exponential form:
z = 2.4456891 × ei -0.4581489 = 2.4456891 × ei (-7π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.4581489 rad = -26.25° = -26°15' = -0.1458333π = -7π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.1934719-1.0817006i
Real part: x = Re z = 2.193
Imaginary part: y = Im z = -1.08170062

#### z15 = ((-5i)^(1/8))*(8^(1/3)) = 2.3158967+0.786141i = 2.4456891 × ei 0.3272492 = 2.4456891 × ei 0.1041667 πCalculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-9π/16) = -0.2385651-1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * (-1+1.7320508i) = -0.2385651361 * (-1) + (-0.2385651361) * 1.7320508076i + (-1.19934793i) * (-1) + (-1.19934793i) * 1.7320508076i = 0.23856514-0.41320694i+1.19934793i-2.07733155i2 = 0.23856514-0.41320694i+1.19934793i+2.07733155 = 0.23856514 + 2.07733155 +i(-0.41320694 + 1.19934793) = 2.3158967+0.786141i
alternative steps
1.2228445 × ei (-9π/16) × 2 × ei 2π/3 = 1.2228445 × 2 × ei ((-9π/16)+2π/3) = 2.4456891 × ei 0.3272492 = 2.4456891 × ei 0.1041667 π =
The result z15
Rectangular form:
z = 2.3158967+0.786141i

Angle notation (phasor):
z = 2.4456891 ∠ 18°45'

Polar form:
z = 2.4456891 × (cos 18°45' + i sin 18°45')

Exponential form:
z = 2.4456891 × ei 0.3272492 = 2.4456891 × ei 0.1041667 π

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.3272492 rad = 18.75° = 18°45' = 0.1041667π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.3158967+0.786141i
Real part: x = Re z = 2.316
Imaginary part: y = Im z = 0.78614099

#### z16 = ((-5i)^(1/8))*(8^(1/3)) = 1.0817006+2.1934719i = 2.4456891 × ei 17π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-5π/16) = 0.679376-1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1+1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * (-1+1.7320508i) = 0.679376028827 * (-1) + 0.679376028827 * 1.7320508076i + (-1.01675807982i) * (-1) + (-1.01675807982i) * 1.7320508076i = -0.67937603+1.1767138i+1.01675808i-1.76107665i2 = -0.67937603+1.1767138i+1.01675808i+1.76107665 = -0.67937603 + 1.76107665 +i(1.1767138 + 1.01675808) = 1.0817006+2.1934719i
alternative steps
1.2228445 × ei (-5π/16) × 2 × ei 2π/3 = 1.2228445 × 2 × ei ((-5π/16)+2π/3) = 2.4456891 × ei 17π/48 =
The result z16
Rectangular form:
z = 1.0817006+2.1934719i

Angle notation (phasor):
z = 2.4456891 ∠ 63°45'

Polar form:
z = 2.4456891 × (cos 63°45' + i sin 63°45')

Exponential form:
z = 2.4456891 × ei 1.1126474 = 2.4456891 × ei 17π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.1126474 rad = 63.75° = 63°45' = 0.3541667π = 17π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.0817006+2.1934719i
Real part: x = Re z = 1.082
Imaginary part: y = Im z = 2.19347188

#### z17 = ((-5i)^(1/8))*(8^(1/3)) = -1.6125549-1.8387664i = 2.4456891 × ei (-35π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei -0.1963495 = 1.2228445 × ei (-0.0625) π = 1.1993479-0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.1993479-0.2385651i) * (-1-1.7320508i) = 1.19934793 * (-1) + 1.19934793 * (-1.7320508076i) + (-0.2385651361i) * (-1) + (-0.2385651361i) * (-1.7320508076i) = -1.19934793-2.07733155i+0.23856514i+0.41320694i2 = -1.19934793-2.07733155i+0.23856514i-0.41320694 = -1.19934793 - 0.41320694 +i(-2.07733155 + 0.23856514) = -1.6125549-1.8387664i
alternative steps
1.2228445 × ei -0.1963495 = 1.2228445 × ei (-0.0625) π × 2 × ei (-2π/3) = 1.2228445 × 2 × ei ((-0.0625)+(-2π/3)) = 2.4456891 × ei (-35π/48) =
The result z17
Rectangular form:
z = -1.6125549-1.8387664i

Angle notation (phasor):
z = 2.4456891 ∠ -131°15'

Polar form:
z = 2.4456891 × (cos (-131°15') + i sin (-131°15'))

Exponential form:
z = 2.4456891 × ei -2.2907446 = 2.4456891 × ei (-35π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -2.2907446 rad = -131.25° = -131°15' = -0.7291667π = -35π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.6125549-1.8387664i
Real part: x = Re z = -1.613
Imaginary part: y = Im z = -1.83876641

#### z18 = ((-5i)^(1/8))*(8^(1/3)) = 0.1599557-2.4404527i = 2.4456891 × ei (-23π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 3π/16 = 1.0167581+0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.0167581+0.679376i) * (-1-1.7320508i) = 1.01675807982 * (-1) + 1.01675807982 * (-1.7320508076i) + 0.679376028827i * (-1) + 0.679376028827i * (-1.7320508076i) = -1.01675808-1.76107665i-0.67937603i-1.1767138i2 = -1.01675808-1.76107665i-0.67937603i+1.1767138 = -1.01675808 + 1.1767138 +i(-1.76107665 - 0.67937603) = 0.1599557-2.4404527i
alternative steps
1.2228445 × ei 3π/16 × 2 × ei (-2π/3) = 1.2228445 × 2 × ei (3π/16+(-2π/3)) = 2.4456891 × ei (-23π/48) =
The result z18
Rectangular form:
z = 0.1599557-2.4404527i

Angle notation (phasor):
z = 2.4456891 ∠ -86°15'

Polar form:
z = 2.4456891 × (cos (-86°15') + i sin (-86°15'))

Exponential form:
z = 2.4456891 × ei -1.5053465 = 2.4456891 × ei (-23π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -1.5053465 rad = -86.25° = -86°15' = -0.4791667π = -23π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.1599557-2.4404527i
Real part: x = Re z = 0.16
Imaginary part: y = Im z = -2.44045268

#### z19 = ((-5i)^(1/8))*(8^(1/3)) = 1.8387664-1.6125549i = 2.4456891 × ei (-11π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 7π/16 = 0.2385651+1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.2385651+1.1993479i) * (-1-1.7320508i) = 0.2385651361 * (-1) + 0.2385651361 * (-1.7320508076i) + 1.19934793i * (-1) + 1.19934793i * (-1.7320508076i) = -0.23856514-0.41320694i-1.19934793i-2.07733155i2 = -0.23856514-0.41320694i-1.19934793i+2.07733155 = -0.23856514 + 2.07733155 +i(-0.41320694 - 1.19934793) = 1.8387664-1.6125549i
alternative steps
1.2228445 × ei 7π/16 × 2 × ei (-2π/3) = 1.2228445 × 2 × ei (7π/16+(-2π/3)) = 2.4456891 × ei (-11π/48) =
The result z19
Rectangular form:
z = 1.8387664-1.6125549i

Angle notation (phasor):
z = 2.4456891 ∠ -41°15'

Polar form:
z = 2.4456891 × (cos (-41°15') + i sin (-41°15'))

Exponential form:
z = 2.4456891 × ei -0.7199483 = 2.4456891 × ei (-11π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -0.7199483 rad = -41.25° = -41°15' = -0.2291667π = -11π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.8387664-1.6125549i
Real part: x = Re z = 1.839
Imaginary part: y = Im z = -1.61255487

#### z20 = ((-5i)^(1/8))*(8^(1/3)) = 2.4404527+0.1599557i = 2.4456891 × ei 0.0654498 = 2.4456891 × ei 0.0208333 πCalculation stepsprincipal root

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 11π/16 = -0.679376+1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.679376+1.0167581i) * (-1-1.7320508i) = -0.679376028827 * (-1) + (-0.679376028827) * (-1.7320508076i) + 1.01675807982i * (-1) + 1.01675807982i * (-1.7320508076i) = 0.67937603+1.1767138i-1.01675808i-1.76107665i2 = 0.67937603+1.1767138i-1.01675808i+1.76107665 = 0.67937603 + 1.76107665 +i(1.1767138 - 1.01675808) = 2.4404527+0.1599557i
alternative steps
1.2228445 × ei 11π/16 × 2 × ei (-2π/3) = 1.2228445 × 2 × ei (11π/16+(-2π/3)) = 2.4456891 × ei 0.0654498 = 2.4456891 × ei 0.0208333 π =
The result z20
Rectangular form:
z = 2.4404527+0.1599557i

Angle notation (phasor):
z = 2.4456891 ∠ 3°45'

Polar form:
z = 2.4456891 × (cos 3°45' + i sin 3°45')

Exponential form:
z = 2.4456891 × ei 0.0654498 = 2.4456891 × ei 0.0208333 π

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.0654498 rad = 3.75° = 3°45' = 0.0208333π rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.4404527+0.1599557i
Real part: x = Re z = 2.44
Imaginary part: y = Im z = 0.15995572

#### z21 = ((-5i)^(1/8))*(8^(1/3)) = 1.6125549+1.8387664i = 2.4456891 × ei 13π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei 15π/16 = -1.1993479+0.2385651i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.1993479+0.2385651i) * (-1-1.7320508i) = -1.19934793 * (-1) + (-1.19934793) * (-1.7320508076i) + 0.2385651361i * (-1) + 0.2385651361i * (-1.7320508076i) = 1.19934793+2.07733155i-0.23856514i-0.41320694i2 = 1.19934793+2.07733155i-0.23856514i+0.41320694 = 1.19934793 + 0.41320694 +i(2.07733155 - 0.23856514) = 1.6125549+1.8387664i
alternative steps
1.2228445 × ei 15π/16 × 2 × ei (-2π/3) = 1.2228445 × 2 × ei (15π/16+(-2π/3)) = 2.4456891 × ei 13π/48 =
The result z21
Rectangular form:
z = 1.6125549+1.8387664i

Angle notation (phasor):
z = 2.4456891 ∠ 48°45'

Polar form:
z = 2.4456891 × (cos 48°45' + i sin 48°45')

Exponential form:
z = 2.4456891 × ei 0.850848 = 2.4456891 × ei 13π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 0.850848 rad = 48.75° = 48°45' = 0.2708333π = 13π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.6125549+1.8387664i
Real part: x = Re z = 1.613
Imaginary part: y = Im z = 1.83876641

#### z22 = ((-5i)^(1/8))*(8^(1/3)) = -0.1599557+2.4404527i = 2.4456891 × ei 25π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-13π/16) = -1.0167581-0.679376i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.0167581-0.679376i) * (-1-1.7320508i) = -1.01675807982 * (-1) + (-1.01675807982) * (-1.7320508076i) + (-0.679376028827i) * (-1) + (-0.679376028827i) * (-1.7320508076i) = 1.01675808+1.76107665i+0.67937603i+1.1767138i2 = 1.01675808+1.76107665i+0.67937603i-1.1767138 = 1.01675808 - 1.1767138 +i(1.76107665 + 0.67937603) = -0.1599557+2.4404527i
alternative steps
1.2228445 × ei (-13π/16) × 2 × ei (-2π/3) = 1.2228445 × 2 × ei ((-13π/16)+(-2π/3)) = 2.4456891 × ei 25π/48 =
The result z22
Rectangular form:
z = -0.1599557+2.4404527i

Angle notation (phasor):
z = 2.4456891 ∠ 93°45'

Polar form:
z = 2.4456891 × (cos 93°45' + i sin 93°45')

Exponential form:
z = 2.4456891 × ei 1.6362462 = 2.4456891 × ei 25π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 1.6362462 rad = 93.75° = 93°45' = 0.5208333π = 25π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.1599557+2.4404527i
Real part: x = Re z = -0.16
Imaginary part: y = Im z = 2.44045268

#### z23 = ((-5i)^(1/8))*(8^(1/3)) = -1.8387664+1.6125549i = 2.4456891 × ei 37π/48Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-9π/16) = -0.2385651-1.1993479i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.2385651-1.1993479i) * (-1-1.7320508i) = -0.2385651361 * (-1) + (-0.2385651361) * (-1.7320508076i) + (-1.19934793i) * (-1) + (-1.19934793i) * (-1.7320508076i) = 0.23856514+0.41320694i+1.19934793i+2.07733155i2 = 0.23856514+0.41320694i+1.19934793i-2.07733155 = 0.23856514 - 2.07733155 +i(0.41320694 + 1.19934793) = -1.8387664+1.6125549i
alternative steps
1.2228445 × ei (-9π/16) × 2 × ei (-2π/3) = 1.2228445 × 2 × ei ((-9π/16)+(-2π/3)) = 2.4456891 × ei 37π/48 =
The result z23
Rectangular form:
z = -1.8387664+1.6125549i

Angle notation (phasor):
z = 2.4456891 ∠ 138°45'

Polar form:
z = 2.4456891 × (cos 138°45' + i sin 138°45')

Exponential form:
z = 2.4456891 × ei 2.4216443 = 2.4456891 × ei 37π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 2.4216443 rad = 138.75° = 138°45' = 0.7708333π = 37π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.8387664+1.6125549i
Real part: x = Re z = -1.839
Imaginary part: y = Im z = 1.61255487

#### z24 = ((-5i)^(1/8))*(8^(1/3)) = -2.4404527-0.1599557i = 2.4456891 × ei (-47π/48)Calculation steps

1. Divide: 1 / 8 = 0.125
2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.2228445 × ei (-5π/16) = 0.679376-1.0167581i
3. Divide: 1 / 3 = 0.33333333
4. Cube root: ∛8 = -1-1.7320508i
5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.679376-1.0167581i) * (-1-1.7320508i) = 0.679376028827 * (-1) + 0.679376028827 * (-1.7320508076i) + (-1.01675807982i) * (-1) + (-1.01675807982i) * (-1.7320508076i) = -0.67937603-1.1767138i+1.01675808i+1.76107665i2 = -0.67937603-1.1767138i+1.01675808i-1.76107665 = -0.67937603 - 1.76107665 +i(-1.1767138 + 1.01675808) = -2.4404527-0.1599557i
alternative steps
1.2228445 × ei (-5π/16) × 2 × ei (-2π/3) = 1.2228445 × 2 × ei ((-5π/16)+(-2π/3)) = 2.4456891 × ei (-47π/48) =
The result z24
Rectangular form:
z = -2.4404527-0.1599557i

Angle notation (phasor):
z = 2.4456891 ∠ -176°15'

Polar form:
z = 2.4456891 × (cos (-176°15') + i sin (-176°15'))

Exponential form:
z = 2.4456891 × ei -3.0761428 = 2.4456891 × ei (-47π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -3.0761428 rad = -176.25° = -176°15' = -0.9791667π = -47π/48 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.4404527-0.1599557i
Real part: x = Re z = -2.44
Imaginary part: y = Im z = -0.15995572
This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As an imaginary unit, use i or j (in electrical engineering), which satisfies the basic equation i2 = −1 or j2 = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

## Basic operations with complex numbers

We hope that work with the complex number is quite easy because you can work with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Very simple, add up the real parts (without i) and add up the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

### Subtraction

Again very simple, subtract the real parts and subtract the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i

### Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply i2 = -1.
This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i

### Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), multiply with conjugate c-di:

(c+di)(c-di) = c2+d2

$\frac{a+bi}{c+di}=\frac{\left(a+bi\right)\left(c-di\right)}{\left(c+di\right)\left(c-di\right)}=\frac{ac+bd+i\left(bc-ad\right)}{{c}^{2}+{d}^{2}}=\frac{ac+bd}{{c}^{2}+{d}^{2}}+\frac{bc-ad}{{c}^{2}+{d}^{2}}i$

(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i

### Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142136
|6i| = 6
abs(2+5i) = 5.3851648

### Square root

Square root of complex number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is to go with De Moivre's formula. Our calculator is on edge because the square root is not a well-defined function on a complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.1213203+2.1213203i
sqrt(10-6i) = 3.2910412-0.9115656i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225i
pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i

### Square, power, complex exponentiation

Our calculator can power any complex number to an integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:
${i}^{i}={e}^{-\pi \mathrm{/}2}$
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.03994-9022199.58262i
i^i = 0.2078795764
pow(1+i,3) = -2+2i

### Functions

sqrt
Square Root of a value or expression.
sin
the sine of a value or expression. Autodetect radians/degrees.
cos
the cosine of a value or expression. Autodetect radians/degrees.
tan
tangent of a value or expression. Autodetect radians/degrees.
exp
e (the Euler Constant) raised to the power of a value or expression
pow
Power one complex number to another integer/real/complex number
ln
The natural logarithm of a value or expression
log
The base-10 logarithm of a value or expression
abs or |1+i|
The absolute value of a value or expression
phase
Phase (angle) of a complex number
cis
is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
conj
conjugate of complex number - example: conj(4i+5) = 5-4i

#### Examples:

cube root: cuberoot(1-27i)
roots of Complex Numbers: pow(1+i,1/7)
phase, complex number angle: phase(1+i)
cis form complex numbers: 5*cis(45°)
The polar form of complex numbers: 10L60
complex conjugate calculator: conj(4+5i)
equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i
system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy
De Moivre's theorem - equation: z^4=1
multiplication of three complex numbers: (1+3i)(3+4i)(−5+3i)
Find the product of 3-4i and its conjugate.: (3-4i)*conj(3-4i)
operations with complex numbers: (3-i)^3

## Complex numbers in word problems:

• Let z1=x1+y1i
Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)
• Cis notation
Evaluate multiplication of two complex numbers in cis notation: (6 cis 120°)(4 cis 30°) Write result in cis and Re-Im notation.
• Linear combination of complex
If z1=5+3i and z2=4-2i, write the following in the form a+bi a) 4z1+6z2 b) z1*z2